Armature rotates both sides of mirrored mesh
You need to have the mirror modifier first (on top) in you modifier stack, then the armature modifier.

In the mirror modifier have the Vertex Groups option enabled. This mirrors your vertex groups if they have _R / _L or simmilar endings. The bones on the left in your armature should have the ending _L and the bones on the right _R, so the vertex groups also get these endings.

If you create the armature, while in armature edit mode, you can switch on the X-Axis Mirror option in the tools panel. With this enabled, you can hold down shift while extruding from a middle bone to create two bones: one with the extension _L and mirrored version with the extension _R.